The behavior of sequences of solutions to the Hitchin-Simpson equations

Abstract

The Hitchin-Simpson equations are first-order non-linear equations for a pair consisting of a connection and a Higgs field. In this paper, we study the behavior of sequences of solutions to the Hitchin-Simpson equations on closed K\"ahler manifolds with unbounded $L^2$ norms of the Higgs fields. We prove a compactness result for the connections and renormalized Higgs fields, which generalizes the work of Taubes and Mochizuki. As applications, we prove that every $\mathbb{Z}/2$ harmonic 1-form on a K\"ahler manifold can be deformed into a sequence of solutions to the Hitchin-Simpson equations. Additionally, we solve the generalized Hitchin's WKB problem on any K\"ahler manifold.